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Simplifying (a + bi)(4 + -2i) = 40 Multiply (a + bi) * (4 + -2i) (a(4 + -2i) + bi(4 + -2i)) = 40 ((4 * a + -2i * a) + bi(4 + -2i)) = 40 ((4a + -2ai) + bi(4 + -2i)) = 40 (4a + -2ai + (4 * bi + -2i * bi)) = 40 (4a + -2ai + (4bi + -2bi2)) = 40 (4a + -2ai + 4bi + -2bi2) = 40 Solving 4a + -2ai + 4bi + -2bi2 = 40 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4bi' to each side of the equation. 4a + -2ai + 4bi + -4bi + -2bi2 = 40 + -4bi Combine like terms: 4bi + -4bi = 0 4a + -2ai + 0 + -2bi2 = 40 + -4bi 4a + -2ai + -2bi2 = 40 + -4bi Add '2bi2' to each side of the equation. 4a + -2ai + -2bi2 + 2bi2 = 40 + -4bi + 2bi2 Combine like terms: -2bi2 + 2bi2 = 0 4a + -2ai + 0 = 40 + -4bi + 2bi2 4a + -2ai = 40 + -4bi + 2bi2 Reorder the terms: -40 + 4a + -2ai + 4bi + -2bi2 = 40 + -4bi + 2bi2 + -40 + 4bi + -2bi2 Reorder the terms: -40 + 4a + -2ai + 4bi + -2bi2 = 40 + -40 + -4bi + 4bi + 2bi2 + -2bi2 Combine like terms: 40 + -40 = 0 -40 + 4a + -2ai + 4bi + -2bi2 = 0 + -4bi + 4bi + 2bi2 + -2bi2 -40 + 4a + -2ai + 4bi + -2bi2 = -4bi + 4bi + 2bi2 + -2bi2 Combine like terms: -4bi + 4bi = 0 -40 + 4a + -2ai + 4bi + -2bi2 = 0 + 2bi2 + -2bi2 -40 + 4a + -2ai + 4bi + -2bi2 = 2bi2 + -2bi2 Combine like terms: 2bi2 + -2bi2 = 0 -40 + 4a + -2ai + 4bi + -2bi2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-20 + 2a + -1ai + 2bi + -1bi2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-20 + 2a + -1ai + 2bi + -1bi2)' equal to zero and attempt to solve: Simplifying -20 + 2a + -1ai + 2bi + -1bi2 = 0 Solving -20 + 2a + -1ai + 2bi + -1bi2 = 0 Move all terms containing a to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 2a + -1ai + 2bi + 20 + -1bi2 = 0 + 20 Reorder the terms: -20 + 20 + 2a + -1ai + 2bi + -1bi2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 2a + -1ai + 2bi + -1bi2 = 0 + 20 2a + -1ai + 2bi + -1bi2 = 0 + 20 Combine like terms: 0 + 20 = 20 2a + -1ai + 2bi + -1bi2 = 20 Add '-2bi' to each side of the equation. 2a + -1ai + 2bi + -2bi + -1bi2 = 20 + -2bi Combine like terms: 2bi + -2bi = 0 2a + -1ai + 0 + -1bi2 = 20 + -2bi 2a + -1ai + -1bi2 = 20 + -2bi Add 'bi2' to each side of the equation. 2a + -1ai + -1bi2 + bi2 = 20 + -2bi + bi2 Combine like terms: -1bi2 + bi2 = 0 2a + -1ai + 0 = 20 + -2bi + bi2 2a + -1ai = 20 + -2bi + bi2 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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